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Preventing Representational Rank Collapse in MPNNs by Splitting the Computational Graph

Andreas Roth, Franka Bause, Nils M. Kriege, Thomas Liebig

TL;DR

This work tackles rank collapse in MPNNs by splitting a single graph into multiple edge-relations and performing message passing across a multi-relational graph. The authors formalize the Multi-Relational Split MPNN (MRS-MPNN) framework and instantiate variants such as MRS-GCN, proving that structural independence between node pairs—captured via weighted in-degrees across relations—guarantees linear independence of updated node representations, thus mitigating over-smoothing. They further show how to obtain multiple relations via a partial ordering to create directed acyclic relations, ensuring the presence of structurally independent nodes. Extensive experiments on ZINC, ZINC12k, and heterophilic graph benchmarks demonstrate improved training dynamics and richer representations, at times with modest runtime overhead, and reveal both the potential and the limitations of graph splitting approaches for real-world tasks where task-specific graph splits can be advantageous.

Abstract

The ability of message-passing neural networks (MPNNs) to fit complex functions over graphs is limited as most graph convolutions amplify the same signal across all feature channels, a phenomenon known as rank collapse, and over-smoothing as a special case. Most approaches to mitigate over-smoothing extend common message-passing schemes, e.g., the graph convolutional network, by utilizing residual connections, gating mechanisms, normalization, or regularization techniques. Our work contrarily proposes to directly tackle the cause of this issue by modifying the message-passing scheme and exchanging different types of messages using multi-relational graphs. We identify a sufficient condition to ensure linearly independent node representations. As one instantion, we show that operating on multiple directed acyclic graphs always satisfies our condition and propose to obtain these by defining a strict partial ordering of the nodes. We conduct comprehensive experiments that confirm the benefits of operating on multi-relational graphs to achieve more informative node representations.

Preventing Representational Rank Collapse in MPNNs by Splitting the Computational Graph

TL;DR

This work tackles rank collapse in MPNNs by splitting a single graph into multiple edge-relations and performing message passing across a multi-relational graph. The authors formalize the Multi-Relational Split MPNN (MRS-MPNN) framework and instantiate variants such as MRS-GCN, proving that structural independence between node pairs—captured via weighted in-degrees across relations—guarantees linear independence of updated node representations, thus mitigating over-smoothing. They further show how to obtain multiple relations via a partial ordering to create directed acyclic relations, ensuring the presence of structurally independent nodes. Extensive experiments on ZINC, ZINC12k, and heterophilic graph benchmarks demonstrate improved training dynamics and richer representations, at times with modest runtime overhead, and reveal both the potential and the limitations of graph splitting approaches for real-world tasks where task-specific graph splits can be advantageous.

Abstract

The ability of message-passing neural networks (MPNNs) to fit complex functions over graphs is limited as most graph convolutions amplify the same signal across all feature channels, a phenomenon known as rank collapse, and over-smoothing as a special case. Most approaches to mitigate over-smoothing extend common message-passing schemes, e.g., the graph convolutional network, by utilizing residual connections, gating mechanisms, normalization, or regularization techniques. Our work contrarily proposes to directly tackle the cause of this issue by modifying the message-passing scheme and exchanging different types of messages using multi-relational graphs. We identify a sufficient condition to ensure linearly independent node representations. As one instantion, we show that operating on multiple directed acyclic graphs always satisfies our condition and propose to obtain these by defining a strict partial ordering of the nodes. We conduct comprehensive experiments that confirm the benefits of operating on multi-relational graphs to achieve more informative node representations.
Paper Structure (53 sections, 4 theorems, 12 equations, 4 figures, 16 tables)

This paper contains 53 sections, 4 theorems, 12 equations, 4 figures, 16 tables.

Table of Contents

  1. Splitting MPNNs into Multi-Relational MPNNs
  2. MRS-GCN
  3. Theoretical Properties
  4. Obtaining Multiple Relations using a Partial Ordering
  5. Experiments
  6. Improving the Learning Process
  7. Evaluating Node Orderings
  8. More Informative Node Representations
  9. Additional Experiments
  10. Comparison with State-of-the-Art
  11. Conclusion
  12. Mathematical Details
  13. Proof of Theorem 4.4
  14. Proof of Theorem 4.5
  15. Proof of Corollary 5.1
  16. ...and 38 more sections

Key Result

Theorem 4.4

(Structurally independent nodes produce linearly independent representations.) Let $\mathbf{A}_1,\dots,\mathbf{A}_l\in\mathbb{R}^{n\times n}$ be $l$ matrices with nodes $v_i,v_j$ being structurally independent. Then, are linearly independent for a.e. $\mathbf{W}_1,\dots,\mathbf{W}_l\in\mathbb{R}^{d\times d^\prime}$ with $d,d^\prime > l \geq 1$ and a.e. $\mathbf{X}\in\mathbb{R}^{n\times d}$ with $

Figures (4)

  • Figure 1: Structurally dependent (a) and independent (b) node pairs. Colors indicate different graphs. Numbers beside edges indicate edge weights. SD refers to the structural dependency of nodes based on $d_1$ and $d_2$ (given by Def. \ref{['def:neigh_struc']}).
  • Figure 2: Different computational graphs for $\textrm{MRS-MPNN}_{\textrm{DA}}$.
  • Figure 3: Comparison of the Rank-one distance (ROD). Mean values over $50$ random seeds.
  • Figure 4: Training loss (MAE) during optimization on ZINC. The learning rate and the number of layers are tuned for each MPNN. Mean values over three runs with standard deviations as semi-transparent areas.

Theorems & Definitions (12)

  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Corollary 5.1
  • Definition 5.2
  • Proposition 5.3
  • proof
  • proof
  • ...and 2 more